You are correct that your work income from your NJ-based employer is not subject to NJ income tax, since you are a non-resident of NJ and you never physically worked inside NJ.
In order to obtain a refund of the incorrectly withheld NJ taxes, you must file a non-resident NJ return on which you show the withheld NJ taxes, but allocate zero income to NJ. (My answer assumes that you have no other income from NJ.)
Your income is of course fully taxable by your home state of NY.
The sequence in which you prepare your state returns is important only if you have income taxed by two states, and thus are able to claim an "other state credit" on your home state return. Since you actually owe no taxes to NJ, there is no credit to be claimed in your situation.
Additionally, if you still have the same job, and your employer will not withhold NY taxes for you, then you must make quarterly estimated tax payments to NY.
https://www.tax.ny.gov/pit/estimated_tax/
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
